JEE MAIN - Physics (2021 - 26th February Evening Shift - No. 11)
The length of metallic wire is l1 when tension in it is T1. It is l2 when the tension is T2. The original length of the wire will be :
$${{{T_1}{l_1} - {T_2}{l_2}} \over {{T_2} - {T_1}}}$$
$${{{l_1} + {l_2}} \over 2}$$
$${{{T_2}{l_1} + {T_1}{l_2}} \over {{T_1} + {T_2}}}$$
$${{{T_2}{l_1} - {T_1}{l_2}} \over {{T_2} - {T_1}}}$$
Explanation
Assuming Hooke's law to be valid.
$$T \propto (\Delta l)$$
$$T = k(\Delta l)$$
Let, l0 = natural length (original length)
$$ \Rightarrow T = k(l - {l_0})$$
so, $${T_1} = k({l_1} - {l_0})$$ & $${T_2} = k({l_2} - {l_0})$$
$$ \Rightarrow {{{T_1}} \over {{T_2}}} = {{{l_1} - {l_0}} \over {{l_2} - {l_0}}}$$
$$ \Rightarrow {l_0} = {{{T_2}{l_1} - {T_1}{l_2}} \over {{T_2} - {T_1}}}$$
$$T \propto (\Delta l)$$
$$T = k(\Delta l)$$
Let, l0 = natural length (original length)
$$ \Rightarrow T = k(l - {l_0})$$
so, $${T_1} = k({l_1} - {l_0})$$ & $${T_2} = k({l_2} - {l_0})$$
$$ \Rightarrow {{{T_1}} \over {{T_2}}} = {{{l_1} - {l_0}} \over {{l_2} - {l_0}}}$$
$$ \Rightarrow {l_0} = {{{T_2}{l_1} - {T_1}{l_2}} \over {{T_2} - {T_1}}}$$
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